1220 
w, = Rsinysiny sind | 2, = Rsinw sin y sin Wp sind 
v, = Rsinzsin weos & | «, = R sin w sin x sin W cos O 
2, = Rn K cos w | v, = Rsinw sin y cos p 
x, = Rsin w cos x 
The expression of the line-element then becomes 
A: ds? = — R*[dyz? + sin? (dy? + sin? Wp d9*)] + ctdt*, 
Be les — R[dw* + sin?widy? + sin? y(dy? 4- sin Wd). 
Finally I perform the “stereographic projection”, and at the same 
time I introduce again rectangular coordinates, by the transformations: 
A | 5 
r.— 2R tan iy h = 22 tan tw 
~ 
=rsinw sin d Lh sin y sin W sin d 
= r sin W cos O 
> 
| 
| y == h sin x sin W cos & 
z==r cos WP | =h sin y cos pp 
a ict — h cos y 
hen oel kalk Ek es derek ad hed hy 
It need hardly be pointed out that in Aw, y,z andin Ba, y, z, ict 
can be arbitrarily interchanged. I put further 
1 
AR? 
The g., for the variables z, y, z, ct then become ') 
i — 
1) In the system B all gu» are infinite on the “hyperboloid” 
1 Joh = 0 oF AR + a? yrs 0) eden 
This discontinuity is however only apparent. The four dimensional world, which 
we have for the sake of symmetry represented as spherical, is in reality hyper- 
bolical, and consists of two sheets, which are only connected with each other at 
infinity. The formulae embrace both sheets, but only one of them represents the 
actual universe. The hyperboloid (a) is the limit between the two parts of the 
euclidean space «x, ¥,2,ct corresponding to these two sheets. It is intersected by 
the axis of ¢ at the points cf — + 2 R, the distance of which from the origin is, 
2K 
edt 
in natural measure, f SS „©. The length in natural measure of the 
—oc'l 
0 
a 
: Ne: dx 
half-axis of 2 is, in both systems, | ———— = ark. 
14+ G2’ 
